Logarithms allow multiplication and division operations to be converted to addition and subtraction. They were invented in the early 1600s by John Napier and Henry Briggs as a way to simplify calculations. Logarithms are closely related to exponents - if x = an then equivalently loga x = n. Studying logarithms allows complex multiplications and divisions to be performed using simpler addition and subtraction operations. They remain important in many fields such as science, engineering, economics and acoustics.

1. MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
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Program Name : CE-6I(replace with your program)
Course Name : Solid Waste Management.(subject)
Academic Year : 2020-2021
Semester : Sixth
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5. .…….…..TOPICS COVERED IN THIS PROJECT…….…....
 INRODUCTION OF LOGARITHM
 WHO INVENTED LOGARITHM
 WHAT IS LOGARITHM?
 WHY DO WE STUDY LOGARITHM?
 FORMULA’S OF LOGARITHM
 USING LOGARITHM TO SOLVE EQUATION

6. Logarithms appear in all sorts of calculations in engineering and
science, business and economics.
Before the days of calculators they were used to assist in the
process of multiplication by replacing
the operation of multiplication by addition. Similarly, they
enabled the operation of division to
be replaced by subtraction. They remain important in other
ways, one of which is that they
provide the underlying theory of the logarithm function. This
has applications in many fields, for
example, the decibel scale in acoustics

7.  INTRODUCTION OF LOGARITHM
In this unit we are going to be looking at logarithms. However, before
we can deal with logarithms
we need to revise indices. This is because logarithms and indices are
closely related, and in order
to understand logarithms a good knowledge of indices is required.
We know that,
16=24
Here, the number 4 is the power. Sometimes we call it an exponent.
Sometimes we call it an index
. In the expression 2 4
, the number 2 is called the base.
Example;
We know that
64=82
.
In this example 2 is the power, or exponent, or index. The number 8 is
the base
X log 3 = (x−2) log 5
Notice now that the x we are trying to find is no longer in a power.
Multiplying out the brackets
X log 3 =x log 5−2 log 5
Rearrange this equation to get the two terms involving X on one side and the

8. remaining term onthe other side
2 log 5 =x log 5−x log 3
Factorise the right-hand side by extracting the common factor of x.
2log 5 =x (log 5-log 3)
=x log 5/3
Using the laws of logarithms.
And finally
X=2 log 5
Log 5/3
If we wanted, this value can be found from a calculator.

9.  WHO INVENTED LOGARITHM
The history of logarithms is the story of a correspondence (in modern
terms, a group isomorphism between multiplication on the positive real
numbers and addition on the real number line that was formalized in
seventeenth century Europe and was widely used to simplify calculation
until the advent of the digital computer. The Napierian logarithms were
published first in 1614. E. W. Hobson called it "one of the very greatest
scientific discoveries that the world has seen." Henry
Briggs introduced common (base 10) logarithms, which were easier to
use. Tables of logarithms were published in many forms over four
centuries. The idea of logarithms was also used to construct the slide rule,
which became ubiquitous in science and engineering until the 1970s. A
breakthrough generating the natural logarithm was the result of a search
for an expression of area against a rectangular hyperbola, and required the
assimilation of a new function into standard mathematics.

10. John Napier of Merchiston,
nicknamed Marvellous
Merchiston, was a Scottish
landowner known as a
mathematician, physicist, and
astronomer. He was the 8th
Laird of Merchiston. His
Latinized name was Ioannes
Neper. John Napier is best
known as the discoverer of
logarithms.
BORN: 1 February 1550, Merchiston Tower , United Kingdom
DIED: 4 April 1617, Merchiston Tower, United Kingdom
CHILDREN: Joan Napier, Robert Napier, Elizabeth Napier
PARENTS: Archibald Napier, Janet Bothwell
SPOUSE: Elizabeth Stirling (m. 1572–1579)
SIBLINGS: Janet Napier, Francis Napier
GRANDCHILDREN: Archibald Napier, 2nd Lord Napier,

11. WHAT IS LOGARITHM?
Consider the expression
16 = 24
Remember that 2 is the base, and 4 is the power. An alternative, yet
equivalent, way of writing this expression is log2 16 = 4. This is stated as
‘log to base 2 of 16equals 4’. We see that the logarithm is the same as the
power or index in the original expression . It is the base in the original
expression which becomes the base of the logarithm.
The two statements,
16 = 24
log2 16=4
are equivalent statements. If we write either of them, we are automatically
implying the other.
Example
If we write down that
64 = 82
then the equivalent statement using logarithms is log8 16=2.
Example
If we write down that log3 27=3
then the equivalent statement using powers is 33
=27.
So the two sets of statements, one involving powers and one involving
logarithms are equivalent . In the general case we have:
KEY POINTS
If x = an
then equivalently log ax= n

12. Let us develop this a little more. Because 10 = 101
we can write
the equivalent logarithmic form log 1010=1.
Similarly, the logarithmic form of the statement 21
=2 is log 22=1.
In general, or any base a, a=a1
and so log aa
=1.
Key point
Log aa=1
We can see from the Examples above that indices and logarithms
are very closely related. Inthe same way that we have rules or laws
of indices, we have laws of logarithms. These aredeveloped in the
following sections.

13.  Why do we study logarithm?
In order to motivate our study of logarithms, consider the following:
we know that 16 = 24
. We also know that 8 = 23
Suppose that we wanted to multiply 16 by 8.
One way is to carry out the multiplication directly using long-multiplication and
obtain 128.But this could be long and tedious if the numbers were larger than 8 and
16. Can we do this calculation another way using the powers ? Note that
16 × 8 can be written 24
× 23
This equals
27
using the rules of indices which tell us to add the powers 4 and 3 to give the new
power, 7. What was a multiplication sum has been reduced to an addition sum.
Similarly if we wanted to divide 16 by 8:
16 ÷ 8 can be written 24
÷ 23
This equals
21
or simply 2
using the rules of indices which tell us to subtract the powers 4 and 3 to give the
new power, 1.
If we had a look-up table containing powers of 2, it would be straight forward to
look up 27
and obtain 27
= 128 as the result of finding 16 × 8.
Notice that by using the powers, we have changed a multiplication problem into
one involving addition (the addition of the powers, 4 and 3). Historicall , this

14. observation led John Napier (1550-1617) and Henry Briggs (1561-1630) to develop
logarithms as a way of replacing multiplication with addition, and also division
with subtraction.

15.  Using logarithms to solve equations
We can use logarithms to solve equations where the unknown is in the
power.
Suppose we wish to solve the equation 3x
= 5. We can solve this by taking
logarithms of both sides. Whilst logarithms to any base can be used, it is
common practice to use base 10, as these are readily available on your
calculator. So,
Log 3x
= log 5
Now using the laws of logarithms, the left hand side can be rewritten to give
x log 3 = log 5
This is more straightforward. The unknown is no longer in the
power. Straightaway
x = log 5
log 3
If we wanted, this value can be found from a calculator.
EXAMPLE
Solve 3x
= 5x-2
. Again, notice that the unknown appears in the power. Take
logs of both sides.
log3x
= log5x-2